-1) In direct numerical simulations of turbulence (DNS) once often uses hyperviscosity. The usual physical viscosity has the form

dv/dt = .. +nu*d2v/dx2

where v is the velocity and nu is the coefficient of the kinematic viscosity.

the hyperviscosity is an artificial term introduced instead of the above term in the form

dv/dt = .. + a*d2nv/dx2n

where n=1, or 2, 3... and a is the coefficient of the hyperviscosity.

Viscosity in general has the effect of dissipating the energy first in the small scales, but the larger scale are also somewhat affected. The hyperviscosity is introduced to actually dissipate mainly the smallest scale without affecting too much the other scales.

-2) Sometimes when studying shocks, oscillations occur near the shock region. At the shock (discontinuity) the pressure gradient is actually very high. In order to get rid of the oscillations it is convenient to have a large viscosity. However, the viscosity is needed only next to the shock. So one can introduce a viscosity that is proportional to the gradient of the pressure or a power of it.

-3) there are many other kind of artificial viscosity for other flow problems. Some are very simple, like for example to smooth out the variables in the physical space with a Shoeman Filter (this correspond to averaging in space the variables once every N timep step, where N is large). There are also spectral filters, that do cut off high frequencies (oscillations) in the flow.

In most cases the artificial viscosity acts like an additional diffusive term.